Assignment 2

Question 1 (Exercise 3.17)

Suppose a proportion 0.001 of the population (i.e., 0.1%) have a certain disease. A diagnostic test is carried out for the disease: the outcome of the test is either positive or negative. It is known from past experience that the test is 90 per cent reliable, i.e. a person with the disease will test (correctly) positive with probability 0.9, while a person without the disease will test (incorrectly) positive with probability 0.1.

  1. A person is tested for the disease. What is the probability that they test positive?

  2. A person’s test result is positive. What is the probability that they have the disease? Comment on your answer.

Question 2 (Exercise 3.26)

At a party you hear that Leo’s birthday is in an earlier month of the year than Matt’s (denote this event by ‘\(L < M\)’). Then you meet Keanu and wonder “what is the chance that Keanu’s birthday is in an earlier month than Leo’s?”. Suppose that each of the three is equally likely to be born in any month of the year and calculate:

  1. \(\pr{L < M}\); (hint: Use P4, with the partition of events \(L_j\) = Leo’s birthday is in month \(j\), with \(j=1,2,\dots, 12\).)

  2. \(\cpr{L_j}{L < M}\), for \(j = 1,2,\dots,12\);

  3. \(\cpr{K < L}{L < M}\).

hint: For the final part of the question, use the conditional version of the partition theorem i.e., \[ \cpr{A}{C} = \sum_j \pr{A \mid B_j\cap C}\cpr{B_j}{C} \] where the \(B_j\) form a partition; also use that conditionally on \(L_j\), \(K < L\) is independent of \(L<M\).

Question 3 (comparative judgement question)

Write a summary, of around forty words, explaining your current best understanding of what it means for two events to be independent of each other. The best summaries are ones that use key ideas, rather than repeating the original definition.

Next week, you will be asked to look at pairs of these summaries, submitted by your peers, and select the summary which you think best encompasses this idea.